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Optimal deterministic algorithm generation

Author

Listed:
  • Alexander Mitsos

    (RWTH Aachen University)

  • Jaromił Najman

    (RWTH Aachen University)

  • Ioannis G. Kevrekidis

    (Princeton University
    FU Berlin
    Johns Hopkins University)

Abstract

A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters—within this family of algorithms—that encode algorithm design: the computational steps of which the selected algorithms consist. The objective function of the optimization problem encodes the merit function of the algorithm, e.g., the computational cost (possibly also including a cost component for memory requirements) of the algorithm execution. The constraints of the optimization problem ensure convergence of the algorithm, i.e., solution of the problem at hand. The formulation is described prototypically for algorithms used in solving nonlinear equations and in performing unconstrained optimization; the parametrized algorithm family considered is that of monomials in function and derivative evaluation (including negative powers). A prototype implementation in GAMS is provided along with illustrative results demonstrating cases for which well-known algorithms are shown to be optimal. The formulation is a mixed-integer nonlinear program. To overcome the multimodality arising from nonconvexity in the optimization problem, a combination of brute force and general-purpose deterministic global algorithms is employed to guarantee the optimality of the algorithm devised. We then discuss several directions towards which this methodology can be extended, their scope and limitations.

Suggested Citation

  • Alexander Mitsos & Jaromił Najman & Ioannis G. Kevrekidis, 2018. "Optimal deterministic algorithm generation," Journal of Global Optimization, Springer, vol. 71(4), pages 891-913, August.
  • Handle: RePEc:spr:jglopt:v:71:y:2018:i:4:d:10.1007_s10898-018-0611-8
    DOI: 10.1007/s10898-018-0611-8
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    References listed on IDEAS

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    1. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    2. Eligius M. T. Hendrix & Boglárka G.-Tóth, 2010. "Nonlinear Programming algorithms," Springer Optimization and Its Applications, in: Introduction to Nonlinear and Global Optimization, chapter 5, pages 91-136, Springer.
    3. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
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    Cited by:

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